Exam 1 Additional Practice Solutions

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Statistics

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Feb 20, 2024

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Solutions (Exam 1 Additional Practice) 1. Suppose that the observational units in a study are the 21 purchases that I made on amazon.com in the year 2018. Identify each of the following as a categorical variable, a quantitative variable, or not a variable. a) How much did I spend on the purchase? quantitative b) Was the purchase shipped to me or to someone else? categorical c) Do I tend to spend more on purchases sent to others than on purchases sent to me? not a variable (research question) d) Did the purchase include free shipping? categorical e) What was the average price of these purchases? not a variable (summary statistic) 2. The National Retail Federation conducted a national survey of 8526 consumers on September 1-9, 2009. Among the findings reported were that: • 29.6% of those surveyed said that the state of the U.S. economy would affect their Halloween spending plans; • the average amount that the respondents said they expect to spend on Halloween is $56.31. a) Identify the observational units in this study. each consumer b) Identify a quantitative variable in this study. amount spent on Halloween c) Are the numbers 29.6% and $56.31 parameters or statistics? Statistics (as the numbers are based on the sample) 3. In the following scenario describe the population, sample, observational unit, variable(s), parameter of interest, and the statistic which could be used to estimate the parameter. Is (are) the variable(s) quantitative or categorical? “A study is to be done on the social interaction skills of California second graders. A random sample of 50 second graders is taken. A psychologist watches each of the selected children for a period of 3 days, and records the number of positive interactions the child had with other children in his/her class at school.” population: all California second graders sample: 50 California second graders OU: each CA second grader variable: number of positive interactions with other children in class - quantitative parameter: average number of positive interactions for all CA second graders statistic: average number of positive interactions for the sample of 50 CA second graders

4. In a May 2013 survey of a random sample of 2,076 U.S. cell phone owners, the Pew Research Center found that 52% use their cell phones to send and/or receive email. Describe the following, in the context of this study: (a) Observational unit: each U.S. cell phone owner (b) Variable: whether or not use cell phone to send and/or receive email (categorical/binary) (c) Parameter of interest? Proportion of % of all U.S. cell phone owners to use their cell phones to send and/or receive email. (d) Which of the following is the best description of the population of interest in this study? A. The proportion of all U.S. cell phone owners who use their cell phones to send and/or receive email. B. All U.S. cell phone owners who use their cell phones to send and/or receive email. C. All U.S. cell phone owners. D. All U.S. adults. 5. The following graph summarizes data collected on a sample of 432 domestic wines. (a) Identify the observational unit in this study. each domestic wine (b) What variable(s) is/are graphed? Be sure to note the variable type. amount of lead (in parts per billion) - quantitative (c) For these data, how will the mean and median compare? Is one bigger than the other or both are the same? Explain how you know. mean will be larger than median, because distribution is skewed to the right. (d) TRUE or FALSE: More than 50% of the sampled wines have a lead content of less than 50 parts per billion. TRUE, because 300/432 is more than 50%

| 6. Automotive News tracks and reports monthly inventories for both domestic and foreign auto manufacturers. The table below illustrates inventories for the Big Three U.S. manufacturers in November of 1998. Manufacturer Cars Trucks Daimler Chrysler 178,300 381,900 Ford 321,300 467,300 General Motors 550,500 433,900 (a) What percentage of the Big Three’s inventory is cars? 45% (b) What percentage of the inventory in the table above comes from Ford? 33.8% (c) What percentage of the General Motors inventory is trucks? 44.1% (d) Of the car inventory, what percentage comes from Ford? 30.6% (e) What percentage of the inventory is manufactured by Ford and is a truck? 20% (f) If a randomly selected automotive from these is a truck, what’s the probability it is made by Ford? P(Ford) = 0. 364 (g) If a randomly selected automotive from these is made by Ford, what’s the probability it is a truck? P(Truck|Ford) = 0. 593 7. Based on the following statistics, describe the distribution of the data: 𝑥𝑥̅ = 13 . 7; median = 17.9; sd = 3.3 distribution is skewed to the left, mean < median. 8. What is the standard deviation of the following data set: 6, 6, 6, 6, 6, 6, 6? 0 (no variability) 9. Under what circumstances will we have a negative valued standard deviation? never 10. Suppose that among the pre-owned cars available in a large dealership, 80% have air conditioning, 70% have a CD player, and 65% have both. CD no CD total AC 0.65 0.15 0.80 no AC 0.05 0.15 0.20 Total 0.70 0.30 1.0 (a) Determine the proportion of these cars that have neither air conditioning nor a CD player. (As always, show your method of solution.) P(no AC and no CD) = 0.15, see table above (b) Determine the conditional probability that a randomly selected car has a CD player, given that it has air conditioning. P ( CD|AC ) = 0.65/0.80 = 0.8125 (c) Are the events randomly selected car has air conditioning and randomly selected car has a CD player independent? Justify your answer. no, P ( CD|AC ) = 0 . 8125 = P ( CD ) = 0 . 70. Having AC increases the car’s chance of also having a CD player. (d) Are the events randomly selected car has air conditioning and randomly selected car has a CD player disjoint? Explain briefly. No, can have both. P(CD and AC) = 0.65 = 0

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